These notes are mostly inspired from reading Cox & Giuli (“Principles of stellar structure”); some insights are from A.Maeder (“Physics, formation and evolution of rotating stars”) and Kippenhahn & Weigert (“Stellar Structure and Evolution”)
In hydrostatic equilibrium we can define the total pressure scale height, \({H_{\mathrm{P}}}\), as \[-\frac{{{\mathrm d}}\ln {\mathrm{P}}}{{{\mathrm d}}r} \equiv \frac{1}{{H_{\mathrm{P}}}} = \frac{\rho g}{{\mathrm{P}}}\] where \({\mathrm{P}}\) is the total pressure (\({\mathrm{P}}_{gas} + {\mathrm{P}}_{rad}\)). The pressure scale height is a measure of the distance over which the pressure changes by an appreciable fraction of itself.